Minimizing Disjunctive Normal Forms of Pure

First-Order Logic

Timm Lampert

This is a penultimate draft of a paper published in Logic Journal of theIGPL 25.3 2017, p. 325-347

Abstract

In contrast to Hintikka’s enormously complex distributive normal forms of first-order logic, this paper shows how to generate minimized disjunctive normal forms offirst-order logic. An effective algorithm for this purpose is outlined, and the benefitsof using minimized disjunctive normal forms to explain the truth conditions of propo-sitions expressible within pure first-order logic are presented.

keywords. first-order logic, disjunctive normal forms, purification, optimization, Quine-McCluskey algorithm

1 Introduction

Unlike disjunctive normal forms (DNFs) of propositional logic, DNFs of first-order logic(FOLDNFs) have received little attention. Instead, prenex normal forms (PNFs) dominatethe metatheoretic study of decidable and undecidable fragments of first-order logic (FOL),and clausal normal forms, particularly Skolem normal forms based on conjunctive normalforms (CNFs), dominate the computational search for effective FOL algorithms. Comparedwith these established normal forms, FOLDNFs have not been studied in the same amountof detail.

Hintikka’s distributive normal forms of FOL constitute an important exception to thisneglect of FOLDNFs.1 These normal forms correspond to canonical DNFs of propositionallogic (CDNFs). They include an exhaustive and mutually exclusive enumeration of disjuncts.Hintikka’s definition of his distributive normal forms is rather intricate; cf. Hintikka [6, pp.52-54], section 3, and in more detail, Nelte [10, pp. 36-65], chapter 3. For the purpose ofthis paper, it suffices to note that the demand for an exhaustive and mutually exclusiveenumeration of disjuncts necessitates an extremely high complexity. The length of eachdisjunct depends on the lengths of (i) the set of all predicates, (ii) the set of all free individual

1Cf. in particular Hintikka [6], which is based on his dissertation, Hintikka [5], which was supervised byGeorg Henrik von Wright, who paved the way for Hintikka; cf., e.g., Wright [21] and Wright [22]. Ehrenfeuchtand Fraisse (cf., e.g., Ehrenfeucht [3]) as well as Oglesby (cf. Oglesby [12]) also simultaneously worked ondistributive normal forms, independently and with different motivations; cf. also Scott [17].

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symbols (or names), and (iii) the maximal length of sequences of nested quantifiers (= depthd). Even if one considers only formulas of pure FOL without names and functions, only onebinary predicate, and formulas of depth 2, this leads to FOLDNFs with 2512 disjuncts, whereeach disjunct contains 512 conjuncts. Thus, the length of Hintikka’s distributive normal formfor even the simple formula ∃x∃yFxy is 2512. Merely increasing the depth by one already

results in 221+235

possible disjuncts (cf. Nelte [10], section 4.1, who provides detailed formulasfor measuring the size of Hintikka’s normal forms). Because of this immense complexity,Hintikka’s normal forms cannot earn more than theoretical interest.

The reason for this complexity lies inHintikka’s intention to use his distributive normalforms in a logical theory of probability and, consequently, in information and game theory.2

However, this motivation faces severe problems in regards to FOL.The application of probability to CDNFs is clear-cut in the case of propositional logic:

Here, any literal has an a priori probability of 12 , and the probabilities of disjuncts and dis-

junctions can be straightforwardly computed by multiplying and adding the correspondingprobabilities. However, things become complicated as soon as FOLDNFs of polyadic FOLare considered. One can no longer presume that conjuncts are logically independent. Hence,it is unclear how to assign probabilities to single conjuncts in Hintikka’s normal forms. Fur-thermore, conjuncts that are logically implied by other conjuncts cannot contribute to theprobability of a disjunct, and disjuncts that are inconsistent cannot contribute to the prob-ability of a disjunction.3 Thus, the application of probability theory in this case is not asclear-cut as in propositional logic, and FOLDNFs cannot be used to compute probabilitiesgiven the undecidability of FOL.

However, the interest in FOLDNFs need not be motivated by a logical theory of probabil-ity. Hintikka was influenced by von Wright, who pioneered the study of distributive normalforms and supervised Hintikka’s dissertation on the topic. Both, in turn, were inspiredby Wittgenstein’s programmatic ideas on logic in general as well as on a logical theory ofprobability based on CDNFs in particular. From a Wittgensteinian perspective, one shoulddistinguish a general motivation and a more specific motivation for studying DNFs. First ofall, DNFs are a perspicuous way to represent the truth conditions of instances of formulas.DNFs in general provide an alternative to the mathematical model-theoretic approach tosemantics. As a consequence of using DNFs to explain the truth conditions of propositions,there is no need to refer to models that rely on set theory and refer to countably or evenuncountably infinite domains. I explain this statement in more detail in section 3. However,although von Wright’s and Hintikka’s study of DNFs is obviously influenced by this gen-eral motivation, it is only their specific interest in a logical foundation of probability thatnecessitates an exhaustive and mutually exclusive enumeration of disjuncts, which, in turn,induces the enormous complexity of their, in a sense, “canonical” disjunctive normal formsof FOL.

Contrary to von Wright and Hintikka, one should seek FOLDNFs of minimal length if oneis interested in representing the truth conditions of instances of formulas by using DNFs as aperspicuous form of knowledge representation. In fact, there is an alternative, much simplerpath leading to FOLDNFs that calls for anti-prenex normal forms that may be written in

2Cf. Hintikka [8] in particular; cf. also Hintikka [7].3Nelte [10], section 4.4, defines a lower bound on the fraction of inconsistent disjuncts. Roughly speaking,

“nearly all” (p. 86) disjuncts of Hintikka’s normal forms are inconsistent as soon as polyadic predicates andoverlapping scopes of quantifiers are considered.

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terms of CNFs or DNFs. However, this path has, to my knowledge, been neither pursuedto generate minimized FOLDNFs nor motivated as an alternative to a traditional semanticapproach to FOL. Behmann [1] is known to be the first to have considered anti-prenexnormal forms to define a decision procedure for monadic FOL. He also called for extendinghis investigations to the entire realm of FOL; see Behmann [1, p. 226]. Church [2, p.58] labels anti-prenex normal forms of monadic FOL “Behmann’s normal forms”. Hilbert &Bernays [4, p. 145ff.] refer to the process of generating Behmann’s normal forms as “analysisto primary formulas”. However, although their description of the process is more explicitthan Behmann’s explanation, they still do not explicitly extend this process to the entirerealm of FOL. This is done in the fourth edition of Quine’s Methods of Logic; see Quine [15],chapter 23. Quine calls the process of generating anti-prenex normal forms “purification”and calls the resulting “primary formulas” “purified formulas”.4 In the following, I willadopt the term “primary formula” from Hilbert and Bernays to refer to conjuncts withindisjuncts of FOLDNFs based on anti-prenex normal forms.

In contrast to Hintikka’s conjuncts of the disjuncts of his distributive normal forms,primary formulas are not required to satisfy certain combinatoric constraints that dependon the set of predicates as well as on the depth of nested quantifiers and thus induce com-plexity. Instead, primary formulas can be defined by conditions concerning the quantifierspreceding conjunctions and disjunctions. These conditions constitute a limit on the drivingof quantifiers inwards to a maximal extent by means of logical equivalence rules that donot yet involve rules for minimizing formulas. For simplicity, throughout the remainder ofthis paper, I avoid names from the language of logic and thus restrict the consideration ofFOL to that of pure FOL. Consequently, all positions in propositional functions of length≥ 1 are occupied by variables bound by universal or existential quantifiers. Neither Hilbert& Bernays [4] nor Quine [15] provides a precise definition of primary formulas. The fol-lowing definition offers a syntactic definition that satisfies the demand to define formulaswith quantifiers that are driven inwards to the maximal extent. This definition is based onnegation normal forms (NNFs), which contain only ∧ and ∨ as dyadic sentential connectivesand ¬ only directly in front of an atomic propositional function.

Definition 1. (Primary formula)

1. Any NNF that does not contain ∧ or ∨ is a primary formula.

2. NNFs that contain ∧ or ∨ are primary formulas iff they satisfy the following conditions:

(a) Any conjunction of n conjuncts (n > 1) is preceded by a sequence of existentialquantifiers of minimal length 1, and all n conjuncts contain each variable of theexistential quantifiers of that sequence.

(b) Any disjunction of n disjuncts (n > 1) is preceded by a sequence of universalquantifiers of minimal length 1, and all n disjuncts contain each variable of theuniversal quantifiers of that sequence.

3. All primary formulas are NNFs that satisfy condition 1 or 2.

Thus, e.g., ∃x∃yFxy and ∃y1(∃y2Fy1y2 ∧ ∃y3∀x1∀x2(Gx1x2y1 ∨ ∀x3Hx1x2x3)) are primaryformulas. By contrast, ∃y1∃y2(Fy1y2∧Gy2) or ∀x1∃y1(Fx1∨Gy1) are not primary formulas.

4Quine [13] already uses anti-prenex normal forms but in a manner that is similar to von Wright’s usage.

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> and ⊥ are also considered to be NNFs and thus, according to condition 1, represent aspecial case of primary formulas.

The second conjunct in conditions 2(a) and 2(b) is needed if one intends to refer toformulas with quantifiers driven inwards to the maximal extent. However, as I note on p.7, neither the procedure suggested by Hilbert and Bernays nor that defined by Quine sat-isfies the condition that each of the existential (universal) quantifiers in the correspondingsequence of quantifiers binds a variable contained in all conjuncts (disjuncts) of the cor-responding conjunction (disjunction). Instead, they only require that this be the case forthe innermost quantifier of a sequence of existential (universal) quantifiers. However, I willshow in the next section how this deficiency can easily be overcome by supplementing theirprocedures with certain rules for the ordering of quantifiers and their scopes.

Based on Definition 1, FOLDNFs, as used in the following, are defined as follows:

Definition 2. (FOLDNF)FOLDNFs are disjunctions of conjunctions of primary formulas.

Both disjunctions and conjunctions may have length 1. Thus, the formula ∃x∃yFxy dis-cussed above is an FOLDNF, although to satisfy Hintikka’s conditions, it must be convertedinto a disjunction with 2512 disjuncts, each containing 512 conjuncts.

Hilbert, Bernays and Quine follow Behmann in considering anti-prenex normal formsessentially in connection with effective decision procedures. Anti-prenex normal forms aretruth functions of primary formulas. They need not be further converted into FOLDNFs.In connection with clausal normal forms, for example, the process of “analysis to primaryformulas”, or “purification” in general, is also known as “miniscoping”, cf. Nonnengart& Weidenbach [11]. Although studying anti-prenex normal forms in terms of FOLDNFsis also of interest for investigating decision problems, I abstain from addressing this issuein the following. Instead, it seems to me surprising that the most obvious and intuitiveuse of DNFs, namely, to explain the conditions for the truth of instances of formulas byfinite means and based on an equivalence procedure within logic alone, is not considered ingreater detail in connection with the motivation to extend the DNFs of propositional logicto FOLDNFs. Unfortunately, von Wright’s and Hintikka’s intent is to use FOLDNFs incombination with probability theory, and thus, they do not investigate the far less complexFOLDNFs expressed in terms of DNFs of primary formulas. In the following, I intend toovercome this deficiency.

In contrast to the metatheoretic use of PNFs and the computational use of clausal normalforms (or, likewise, anti-prenex normal forms), FOLDNFs expressed in terms of disjunctionsof conjunctions of primary formulas can be utilized for the logico-philosophical enterprise ofanalysing and explaining first-order formulas (or their instances) by converting the initialformulas, mechanically and through equivalence transformation, into a more perspicuousform that allows one to identify the truth conditions of instances of the initial formulas in aplain and neat way. This goal is important for a perspicuous means of knowledge represen-tation based on the language of FOL. If one wishes to understand first-order formulas, onemust answer the question of what they contribute to the truth conditions of their instancesby virtue of their logical form. The best way to answer this question is by referring to theirminimized FOLDNFs. Before I address the question of minimizing FOLDNFs in section4, I elucidate how to explain truth conditions by means of FOLDNFs in section 3. First,

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however, let us specify the process of purification.

2 Purification

In this section, I describe an algorithm that (i) drives quantifiers inwards to the greatestpossible extent and (ii) expands the scopes of the quantifiers to the minimum possible extentwhile achieving (i).

The process of purification begins from NNFs of FOL. If an initial formula is not anNNF of this sort, it can be converted into such a form in accordance with well-known rules.

Let our initial formula φ be

∃y∀x(¬(((¬Fxx ∧Gy) ∨Hxy)→ ¬P )) (1)

Then, the process of purification starts with the NNF of (1):

∃y∀x(((¬Fxx ∧Gy) ∨Hxy) ∧ P ) (2)

In contrast to prenexing, purification drives quantifiers inwards. This is done by applyingPN rules, i.e., the rules used for prenexing, in the opposite direction (= miniscoping). Table1 lists the PN laws. They are applied from left to right to drive quantifiers inwards.

∀ν(A ∧ B(ν)) a` A ∧ ∀νB(ν) PN1∀ν(B(ν) ∧ A) a` ∀νB(ν) ∧ A PN2∀ν(A ∨ B(ν)) a` A ∨ ∀νB(ν) PN3∀ν(B(ν) ∨ A) a` ∀νB(ν) ∨ A PN4∃ν(A ∧ B(ν)) a` A ∧ ∃νB(ν) PN5∃ν(B(ν) ∧ A) a` ∃νB(ν) ∧ A PN6∃ν(A ∨ B(ν)) a` A ∨ ∃νB(ν) PN7∃ν(B(ν) ∨ A) a` ∃νB(ν) ∨ A PN8∀ν(A(ν) ∧ B(ν)) a` ∀νA(ν) ∧ ∀νB(ν) PN9∃ν(A(ν) ∨B(ν)) a` ∃νA(ν) ∨ ∃νB(ν) PN10

Table 1: PN laws applied to drive quantifiers inwards

First, universal quantifiers preceding a conjunction and existential quantifiers precedinga disjunction are driven inwards to the maximal extent by applying PN1 to PN10. (= 1stminiscoping).

In the case of formula (2), this results in the following formula:

∃y∀x((¬Fxx ∧Gy) ∨Hxy) ∧ P (3)

To drive quantifiers farther inwards, the quantifiers must first be sorted (= quantifiersorting). This is done by first generating sequences of universal (existential) quantifiers of

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maximal length that precede disjunctions (conjunctions): All universal (existential) quanti-fiers that are separated only by disjunctions (conjunctions) are grouped together (= partialprenexing). This is done by applying PN3 to PN6 from right to left. Renaming of the vari-ables ensures that no two quantifiers bind the same variable.5 Then, sequences of universal(existential) quantifiers preceding a disjunction (conjunction) are sorted such that universal(existential) quantifiers binding variables that occur in m of the disjuncts (conjuncts) ap-pear to the right of universal (existential) quantifiers binding variables that occur in n ofthe disjuncts (conjuncts), where m < n (= prenex sorting). This is done by applying thefollowing laws:

∀µ∀νA(µ, ν) a` ∀ν∀µA(µ, ν) ∀V∃µ∃νA(µ, ν) a` ∃ν∃µA(µ, ν) ∃V

Table 2: Laws applied for quantifier sorting

Likewise, the disjuncts are sorted such that disjuncts containing a variable µ bound bya universal quantifier ∀µ that appears to the right of a universal quantifier ∀ν in the samesequence of universal quantifiers also appear to the right of disjuncts containing ν (andnot µ). Thus, if ∀µ is the innermost quantifier of a sequence of universal quantifiers, thenall disjuncts containing µ are selected and placed farthest to the right in the order of thedisjuncts in the scope of universal quantifiers, and so on for all other universal quantifiers(= scope sorting). The same process is applied to the scopes of each sequence of existentialquantifiers with respect to the order of the conjuncts. This process involves the applicationof well-known rules of propositional logic:

A ∧B a` B ∧A KOM∧A ∨B a` B ∨A KOM∨(A ∧B) ∧ C a` A ∧ (B ∧ C) ASS∧(A ∨B) ∨ C a` A ∨ (B ∨ C) ASS∨Table 3: Laws applied for scope sorting

On the basis of this sorting, universal quantifiers preceding disjunctions and existentialquantifiers preceding conjunctions can be driven inwards by again applying the PN lawsfrom left to right (= 2nd miniscoping).

Neither Hilbert & Bernays [4, pp. 144-148] nor Quine [15, pp. 126-129] considers quan-tifier sorting. However, the following simple example demonstrates that this procedure isrequired to drive the quantifiers inwards as far as possible:

∃y2∃y1(∃y3(Hy3 ∧Ky3) ∧ Fy1y2 ∧Gy1) (4)

Without quantifier sorting, the scopes of ∃y1 and ∃y2 are not minimized. Thus, according tothe algorithm described by Hilbert, Bernays and Quine, (4) is a primary formula, althoughit does not fully satisfy condition 2 of Definition 1. However, quantifier and scope sorting

5I omit the details of the necessary renaming techniques in the following; for details, cf. Nonnengart &Weidenbach [11, p. 344f.], section 3.4.

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results in the following formula:

∃y3∃y1∃y2(Hy3 ∧Ky3 ∧Gy1 ∧ Fy1y2) (5)

Applying the PN laws to (5) in a 2nd process of miniscoping results in the following:

∃y3(Hy3 ∧Ky3 ∧ ∃y1(Gy1 ∧ ∃y2Fy1y2) (6)

In the case of formula (3), however, no sorting of quantifiers or scopes is needed, as thesequences of existential or universal quantifiers are all of length 1. Consequently, no 2ndminiscoping is applied.

The two miniscoping processes mentioned thus far minimize the scopes of the quantifierswithout expanding their scopes a priori. However, for the scopes of the quantifiers to beminimized to the greatest possible extent, the scopes of all universal quantifiers ∀ν must beconverted into CNFs if they are of the form A(ν) ∨ B(ν), and the scopes of all existentialquantifiers ∃µ must be converted into DNFs if they are of the form A(µ) ∧ B(µ) (= scopeconversion).6 This is done by applying the distributivity laws of propositional logic:

(A ∨ B) ∧ C a` (A ∧ C) ∨ (B ∧ C) DIS1(A ∧B) ∨ C a` (A ∨ C) ∧ (B ∨ C) DIS2

Table 4: Distributivity laws for converting scopes into CNFs orDNFs

In the case of formula (3), the scope of ∀x must be converted to a CNF by first applyingDIS2:

∃y∀x((¬Fxx ∨Hxy) ∧ (Gy ∨Hxy)) ∧ P (7)

Hilbert & Bernays [4, pp. 144-148] convert the scopes of universal (existential) quantifiersinto CNFs (DNFs) regardless. However, this needlessly expands the scopes of the quantifiers.Instead, one should first apply PN laws to drive the quantifiers inwards as far as possiblewithout converting their scopes into CNFs/DNFs. Only if the stated conditions are satisfiedmust the scopes be converted. Afterwards, the miniscoping process described above isapplied again until the scopes of the quantifiers are no further reduced by this process.

Applying first PN9 and then PN3 to (7) results in

∃y(∀x(¬Fxx ∨Hxy) ∧ (Gy ∨ ∀xHxy)) ∧ P (8)

Reiterating once more the entire process described thus far requires applying DIS1 to convertthe scope of ∃y into a DNF. Applying PN10 subsequent to DIS1 finally results in the followingdisjunction of conjunctions of primary formulas:

∃y(∀x(¬Fxx ∨Hxy) ∧Gy) ∧ P ∨ (9)

∃y(∀xHxy ∧ ∀x(¬Fxx ∨Hxy)) ∧ P6This conversion is not considered in the miniscoping process described by Nonnengart & Weidenbach

[11, p. 343f.]. This is why their miniscoping process (apart from ignoring quantifier sorting) does not resultin formulas with scopes that are minimized to the maximal extent.

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After the purification process is complete, the resulting expression is converted into anFOLDNF by applying DIS1. In the case of (9), no further application of DIS1 is necessary.The entire process of purification as described thus far may result in the replication ofquantifiers. The replicated quantifiers bind variables of the same type. In the final step of thealgorithm, the variables are renamed such that the variables bound by universal quantifiersin a given disjunct of the FOLDNF, in which m universal quantifiers occur, are denoted byx1 . . . xm. Likewise, the variables bound by existential quantifiers are denoted by y1 . . . yn,where n is the number of existential quantifiers that occur in the disjunct. Thus, no variableof a disjunct is bound by more than one quantifier in that disjunct (maximal renaming).For our purposes, it is sufficient to use different variables for different quantifiers withinany disjunct of a FOLDNF. In the case of (9), maximal renaming results in the followingFOLDNF:

∃y1(∀x1(¬Fx1x1 ∨Hx1y1) ∧Gy1) ∧ P ∨ (10)

∃y1(∀x1Hx1y1 ∧ ∀x2(¬Fx2x2 ∨Hx2y1)) ∧ P

When the minimization of an FOLDNF is considered (cf. section 4.2), the variablesare renamed such that the minimum number of different variables are used to identify theinternal relations between primary formulas in any case (minimal renaming).

Thus, it is possible to implement a procedure to convert any formula φ of pure FOL intoa formula relating completely purified formulas using the following algorithm:7

1. Convert φ into an NNF.

2. Apply the 1st miniscoping procedure.

3. Sort the quantifiers.

4. Sort the scopes.

5. Apply the 2nd miniscoping procedure.

6. Convert the scopes.

7. Reiterate steps 2-6 until no further changes occur.

8. Convert the resulting formula into an FOLDNF.

9. Rename the variables of each disjunct in accordance with the principle of maximalrenaming.

The resulting FOLDNF is equivalent to the initial formula φ, as only equivalence rules areapplied. The procedure terminates, as it is well known that conversion into NNFs and DNFsas well as the processes of renaming and sorting terminate. Furthermore, the miniscopingprocesses terminate, as they reduce the depth of a formula starting with a quantifier. Theentire process results in disjunctions of conjunctions of primary formulas as defined above,

7For simplicity, I abstain from a more technical definition of the algorithm and instead refer simply tothe names of the steps of the procedure as introduced in italics above. Aside from the equivalence rules usedin the renaming process (cf. Nonnengart & Weidenbach [11, p. 344f.]), all of the logical equivalence rulesinvolved are mentioned above, and the way in which they are applied in each step is trivial.

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as PN1 to PN10 are applied to drive quantifiers inwards to the maximal extent, which ismade possible by sorting strategies and scope conversions. The only factors limiting thedriving in of quantifiers without applying equivalence laws to minimize formulas are thelack of equivalence rules for the cases of ∀ν(A(ν) ∨ B(ν)) and ∃µ(A(µ) ∧ B(µ)) and thenon-equivalence of changing the quantifier order in the case of ∃µ∀ν or ∀ν∃µ.

Compared with a procedure that yields Hintikka’s distributive normal forms8, the de-scribed algorithm yields far less complex FOLDNFs and is, in fact, quite trivial and effective.The purification procedure (steps 1 to 7) is identical to Quine’s procedure as presented inQuine [15, pp. 126-129] except for steps 3 to 6, which are omitted by Quine. Nonnengart& Weidenbach [11, pp. 341-343] refer only to steps 1 and 2. Hilbert & Bernays [4, p. 145]suggest to begin with PNFs and to minimize each quantifier sequentially from the insideto the outside of the prenex, starting with scope conversion (step 6) followed by miniscop-ing (step 2). However, this procedure satisfies neither condition (i) nor condition (ii) asstated above (cf. p. 5). It expands the scopes to the maximal extent and does not consideroptimized quantifier orders. By contrast, the algorithm described in this section performsiterative optimized applications of PN laws to minimize the scopes to the maximal extentwith the minimum number of necessary scope expansions. Because Hilbert & Bernays [4,pp. 144-148] as well as Quine [15, pp. 126-129] omit steps 3 to 6, they do not satisfythe condition of Definition 1 that any disjunction (conjunction) occurring in a primaryformula must be preceded by universal (existential) quantifiers, all of which bind variablesthat occur in each disjunct (conjunct). This may be the reason why Hilbert & Bernays [4,p. 144] explicitly abstain from considering primary formulas of non-monadic FOL, as theyjudge such formulas to lack a sufficient degree of standardization.

The resulting FOLDNF obtained by applying steps 1 to 9 is a disjunction of conjunctionsof purified (or primary) formulas satisfying Definition 2. The primary formulas of FOLD-NFs are the equivalent of the literals in the DNFs of propositional logic. In the following,I first consider the purpose of such FOLDNFs in general (section 3) and then propose aprocedure for minimizing them (section 4).

3 Explaining Truth Conditions

The outward form of arbitrary FOL formulas does not make it possible to identify or readoff the truth conditions of their instances. This becomes clear when one considers theirliteral paraphrases: In most cases, such a paraphrase does not help to understand truthconditions. This situation changes immediately when one considers DNFs. Each disjunctof a DNF identifies a sufficient condition for truth within a logical space of possible truthconditions; the truth of at least one disjunct is a necessary condition for the truth of aninstance of the initial formula. Each disjunct, in turn, enumerates the conditions for truththat must collectively be satisfied for the truth of the disjunct. In paraphrasing DNFs, it isnot necessary to paraphrase logical constants literally. One can think of a disjunction as anenumeration of sufficient conditions for the truth of an instance of the initial formula andof a conjunction as an enumeration of necessary conditions for the truth of an instance of

8Hintikka was satisfied simply to have proven the existence of his distributive normal forms. Nelte [10,p. 81f.], however, describes an algorithm for converting an arbitrary formula φ into its Hintikka distributivenormal form.

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a disjunct. The negation sign occurs only to the left of atomic expressions (propositionalfunctions) and may be paraphrased as indicating that certain atomic conditions are false (ornot satisfied). Likewise, a lack of negation of certain atomic expressions indicates that thecorresponding atomic conditions are true (satisfied). In this respect, a paraphrase of DNFs inthe form of paraphrasing truth conditions serves as an analysans or explanans of an instanceof a logical formula, this instance being the analysandum or explanandum. Consequently, theprocess of transforming formulas into FOLDNFs is a mechanical equivalence transformationwithin a process of analysis or explanation. The explanatory power of such an explanationarises from the resulting form of the explanans, which allows one to identify truth conditions.

Although the pure form of FOLDNFs already contributes to the identification of truthconditions, several questions and problems remain. Let us, for now, put aside intricatequestions concerning internal relations between components of FOLDNFs that induce re-dundancies and motivate minimization. I address these questions in section 4. For simplicity,let us also, for now, exclude internal relations within disjuncts by allowing each predicateto occur only once in a given disjunct of an FOLDNF. In this case, only internal relationsbetween disjuncts may arise, which we also will not address for the time being. We still needto clarify how primary formulas contribute to the identification of conditions for truth. Letus do so by considering the following example.

Let our initial formula φ be formula (1), p. 5:

∃y∀x(¬(((¬Fxx ∧Gy) ∨Hxy)→ ¬P )) (1)

It is obvious that a literal paraphrase that verbalizes the logical constants and the wayin which they are connected is not of much help for understanding the truth conditions ofthis formula.

φ is converted into the FOLDNF given in formula (9), p. 7, according to the algorithmdescribed in section 2. Formula (9), in turn, can be minimized to the following equivalentFOLDNF, composed of the two disjuncts (11) and (12):

∃y1∀x1Hx1y1 ∧ P ∨ (11)

∃y1(∀x1(¬Fx1x1 ∨Hx1y1) ∧Gy1) ∧ P (12)

Given this syntax, conditions for truth can be directly read off in a standardized way.Before explaining the paraphrase of (11)∨(12) in more detail, I note that from here on, I willabstain from cumbersome references to “instances” of formulas and simply speak of “truthconditions of formulas”, tacitly meaning the truth conditions of their instances. As we areconcerned with formulas and their contribution to the identification of truth conditions, thissimplified terminology suffices for our purposes. Thus, we obtain the following paraphraseof (11) ∨ (12):

φ is true iff

– - Some object in the second position of Hx1y1 combined with all objects in thefirst position of Hx1y1 makes the dyadic propositional function Hx1y1 true, and

- the atomic proposition P is true, or

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– - Some object, the same in the second position of Hx1y1 and in the first positionof Gy1, combined with all objects distributed among (i) the first and secondpositions of Fx1x1 (where the same object satisfies these two positions) and (ii)the first position of Hx1y1, makes the dyadic propositional function Fx1x1 false,the dyadic propositional function Hx1y1 true, and the monadic propositionalfunction Gy1 true, and

- the atomic proposition P is true.

In general, an existential quantifier within a primary formula indicates that one andthe same object satisfies all positions of the propositional functions in which the variablethat it binds occurs. In the case of universal quantifiers, one must distinguish whethertheir positions occur in propositional functions that are related by disjunctions or not.In the first case, a universal quantifier indicates that all objects are distributed among thecorresponding positions; in the second case, a universal quantifier indicates that all of a set ofdistributed objects satisfy the corresponding positions. Furthermore, sequences of quantifiersrefer to specific combinations of objects that satisfy certain conditions. Within a sequenceof quantifiers, the order of the existential and universal quantifiers is significant. The orderof existential (universal) quantifiers within sequences of existential (universal) quantifiers,however, is not significant. The same applies to quantifiers that occur in different sequences,as the quantifiers of one sequence are not within the scope of a quantifier of another. Allthese rules for interpreting primary formulas presume their standardized syntax. Unless thesyntax is standardized in this way, it is not possible to establish unambiguous rules thatallow one to read off truth conditions from the syntactic features of FOLDNFs.

Paraphrasing FOLDNFs in this way makes it evident that such paraphrases do not merelyverbalize a logical notation but rather explain the truth conditions of logical formulas. Thiscan be made even more evident by introducing an “iconic notation” for FOLDNFs that doesnot make use of the usual logical constants.9 In this notation, the positions of variables areexplicitly indicated by numbers, and bound variables are replaced with “forks” connectingnumbers, thus explicitly symbolizing the relations of the bound variables to their positions

(cf. table 5). By using an “open fork” (i.e.,/

\) to connect the positions of a universal

variable occurring in different positions of disjunctive connected propositional functions,it is symbolized that the objects of a domain are distributed among those positions. Bycontrast, a closed fork (i.e., <) indicates that the same object must satisfy all positions thatare connected by the fork. Furthermore, T or F indicates that a certain combination of objectsmakes some propositional function true or false (satisfies that function or not), respectively.Finally, disjuncts (conjunctions of length ≥ 1) are expressed as finite sets enumerating theiconic expressions of primary formulas, and a disjunction (of length ≥ 1) is represented by afinite set containing the enumeration of those sets. To such a set representing a disjunction,an outward T is assigned to indicate a set of conditions for truth.10 These conventions makesuperfluous the use of the logical constants ¬,∧ and ∨ as well as the use of variables; they

9Both the idea of an iconic notation for FOL and some of the features of this iconic notation as summarizedabove can be traced back to Peirce’s graphs (cf. Shin [18]), Wittgenstein’s ab-notation (cf. Wittgenstein[20, p. 95f.]), and Quine’s quantificational diagrams using bonds (cf. Quine [16, p. 70]).

10One can easily imagine a complementary set consisting of the conditions for the falsehood of φ. Sucha complementary set can be generated either from the truth conditions of ¬φ or directly from the truthconditions of φ by applying rules for inversion.

11

explain the meaning of the logical constants by reducing the initial formula to its iconicsymbol.

By applying the stated conventions, the FOLDNF (11) ∨ (12) can be translated into itsiconic symbols, from inside to outside, via the following steps:

1. Translate the propositional functions (indicate positions with numbers that replacevariables, and denote affirmation by T and negation by F): Hxy ⇒ T-H12, P ⇒ T-P ,¬Fxx⇒ F-F12, and Gy ⇒ T-G1.

2. Symbolize the relations of the bound variables to their positions. Use forks to connectthe numbers of the positions as follows:

(a) Open forks connect the numbers of positions connected by a disjunction andbound by a universal quantifier,

(b) Closed forks connect the numbers corresponding to all other positions of one andthe same variable.

Thus, ∨ and ∧ as well as the bound variables are eliminated in favour of forks con-necting the positions of the corresponding propositional functions. In the case thata bound variable occurs only once within a propositional function, no fork is needed.This results in the translation of the primary formulas containing quantifiers:

∃y1∀x1Hx1y1 ⇒ ∃2∀1 T-H12

∃y1(∀x1(¬Fx1x1 ∨Hx1y1) ∧Gy1)⇒

< 12

F-F12

∀ rrNNN

N2 1 T-H12

∃ uuuIII1 T-G1

3. Translate the conjunctions of the primary formulas as sets of the translations of theprimary formulas, and translate the disjunction of the conjunctions of the primaryformulas as the set of the translations of the conjunctions. Prepend to the latter anoutward T to indicate a set of truth conditions. This results in the iconic symbol of(11) ∨ (12), i.e., the iconic symbol for the following formula:

∃y1∀x1Hx1y1 ∧ P ∨ ∃y1(∀x1(¬Fx1x1 ∨Hx1y1) ∧Gy1) ∧ P

cf. table 5.

I abstain here from elaborating (i) the rules for translating traditional FOLDNFs intoiconic notation and (ii) the rules for paraphrasing FOLDNFs in traditional or iconic notationin more detail. It is hoped that the iconic notation should speak for itself. Once oneis accustomed to it, there is no need for the cumbersome and less transparent ordinaryparaphrases of FOLDNFs. In contrast to ordinary FOLDNFs, the iconic notation replacesthe ordinary notation of FOL with a notation that allows truth conditions to be identifiedfrom syntactic properties to the greatest possible extent. This is why the iconic notationprovides an explanans for the truth conditions of the initial formula (the explanandum).Because it separates the sufficient and collectively necessary conditions, these conditions

12

T−

{∃2∀1 T-H12, T-P},< 1

2F-F12

∀ rrNNN

N2 1 T-H12 , T-P

∃ uuuIII1 T-G1

Table 5: Iconic symbol for (11) ∨ (12)

being composed of standardized primary expressions, it also provides an analysans of theinitial formula (the analysandum).

The iconic expressions of disjuncts that do not contain any propositional function morethan once can be regarded as providing syntactic criteria for identifying models given asuitable finite domain (an upper bound on the cardinality of such a domain is the maximalnumber of nested quantifiers that may appear within the primary formulas of the corre-sponding disjunct). In the case of the second set appearing in the iconic symbol presentedin table 5, one must bear in mind that models depend on pairs that are not included in=(F ). To make this transparent, one can explicitly enumerate tuples that do not satisfy thecorresponding propositional functions, where such tuples are indicated by an overline. Tocome to understand how models are identified by iconic symbols, one must focus on suchoverlined tuples in the case of propositional functions preceded by “F”. Given such a rep-resentation, one can, for example, easily recognize how the following model, with a domainof two objects, {1, 2}, satisfies the syntactic criteria specified by the second set in the iconicsymbol shown in table 5 by focusing on the underlined tuples:

=(P ): T,

=(F ): {(1, 1), (2, 2), (1, 2), (2, 1)},

=(G): {1, 2},

=(H): {(1, 1), (1, 2), (2, 1), (2, 2)}.

Such a model is more specific than the iconic symbol, as one must arbitrarily choose specificobjects to satisfy the positions of the propositional functions. In this respect, models extendbeyond the language of pure FOL and demand more than is necessary to identify truthconditions. They specify instances that make propositions true rather than the structuralfeatures that instances must satisfy for a proposition to be true.

It is possible to define a mechanical procedure for generating finite models by satisfyingthe conditions coded by the iconic notation for primary formulas of a disjunct that doesnot contain any propositional function more than once.11 However, as this method ingeneral applies only to disjuncts without internally related components and as the followingdiscussion primarily concerns the minimization of FOLDNFs, I abstain from providing the

11In fact, I have implemented an algorithm to generate such models from FOLDNFs, in addition to thealgorithm for generating minimized FOLDNFs described in section 4.3.

13

details here. For now, it is sufficient to mention this use of the logical notation for FOLDNFsto demonstrate the capability of FOLDNFs, and the iconic notation in particular, to enablethe identification of truth conditions based on syntactic criteria. Contrary to a model-theoretic evaluation of a formula, the FOLDNF (11) ∨ (12) as well as the resulting iconicsymbol can be paraphrased from outside to inside, thus identifying the properties that anymodel of the initial formula must satisfy.

These general remarks must suffice to explain how FOLDNFs express the truth conditionsof formulas. There is no comparable means of explaining formulas in terms of enumeratingthe sufficient and collectively necessary conditions for truth. Model theory can do no morethan provide instances of sufficient conditions for truth (or falsehood, in the case of counter-models). It is not based on an equivalence procedure within the realm of logic alone and thusgoes beyond pure FOL. Compared with models, FOLDNFs of pure FOL provide only partialdescriptions, in the sense that they do not mention specific objects. The representation oftruth conditions (or, roughly speaking, of “possible worlds” or “knowledge”) does not extendbeyond the expressive power of the language used for such representation. In particular, itfollows that there is no need to refer to infinite domains when representing truth conditions.12

However, FOLDNFs provide not only a finite but also a neat way to code truth conditions(or represent knowledge). This is particularly true in the case of minimized FOLDNFs (cf.section 4).

Compared with Hintikka’s distributive normal forms, the FOLDNFs generated by thealgorithm described in section 2 are also partial descriptions in another sense: They donot enumerate exhaustive and mutually exclusive sufficient conditions for truth. Instead,they usually provide the sufficient and collectively necessary conditions for truth in a densemanner by ignoring the requirement for mutual exclusivity. This may have disadvantagesfor certain purposes because not all conditions for truth that can be separated are, in fact,explicitly separated. Furthermore, FOLDNFs, even completely minimized ones, do notsatisfy the demand for an unambiguous representation in the sense of providing one and

12At least, this is true as long as one does not claim more than a correct representation of truth conditions.If one additionally claims the ability to generate models from disjuncts of FOLDNFs, it is only true as long asone does not consider disjuncts with propositional functions that occur more than once. This is because of theexistence of disjuncts of FOLDNFs that are satisfied only within an infinite domain, e.g., the transformationof the axiomatic system described by Hilbert & Bernays [4, p. 14] into a conjunction of primary formulas:

∀x1∃y1Fx1y1 ∧ ∀x2∀x3(¬Fx2x3 ∨ ∀x4(¬Fx3x4 ∨ Fx2x4)) ∧ ∀x5¬Fx5x5 (13)

If one further claims the ability to identify internal relations and logical properties from FOLDNFs, onemust claim certain degrees of optimization to dispense with model theory and infinite models. Like modeltheory in the case of infinite domains, the optimization of FOLDNFs faces difficulties related to algorithmicrealization; cf. section 4.1. However, questions concerning the optimization of FOLDNFs still refer not toinfinite domains but rather to the optimization of finite expressions.

14

the same FOLDNF for all formulas of a class of equivalent formulas.13 Satisfying such aclaim within FOL would imply the need for a decision procedure. However, FOLDNFs mayserve various purposes. In addition to the claims of (i) a completely specific representationor (ii) an unambiguous representation, one might also claim that the minimized FOLDNFsprovide the neatest possible coding of truth conditions. Such a claim is very natural, notonly for reasons of economy. It also respects the intuition that a representation of truthconditions should not contain any irrelevant features. Thus, all redundancies should beeliminated. From this consideration, the necessity of an algorithm for minimizing FOLDNFsarises. Such an algorithm becomes of prominent interest as soon as internal relations amongthe components of FOLDNFs are considered, as we do in the next section by discussingminimization strategies for FOLDNFs.

4 Minimization

In propositional logic, an algorithm for minimizing DNFs is readily available: the Quine-McCluskey algorithm. This algorithm is of significant importance in, e.g., engineering,information theory, and causal theory. Thus, a question arises regarding the extent to whichit is possible to generalize such an algorithm. In the following, I first identify the problemswith extending the Quine-McCluskey algorithm to the minimization of FOLDNFs and thendescribe a pragmatic algorithm of my own for minimizing (or optimizing) FOLDNFs.

4.1 Problems with Complete Minimization

For two reasons, a complete minimization algorithm for FOLDNFs is unrealistic: From atheoretical point of view, such an algorithm must presume the decidability of FOL, e.g., toeliminate inconsistent disjuncts. From a practical point of view, such an algorithm wouldhave to be very complex. In propositional logic, all primary formulas (= atomic propositionsand their negations) are consistent and non-tautologous, and they are logically independentof each other apart from their negations. In the general case of FOLDNFs, however, thecomplexity of minimization increases because the primary formulas of FOLDNFs may involveall kinds of logical dependencies:

– primary formulas may be tautologous or inconsistent (atomic inconsistency);

– a conjunction of consistent primary formulas, none of which is the negation of another,may still be inconsistent (implicit inconsistency);

13Within propositional logic, this demand is satisfied by the so-called reductive DNFs (RDNFs) thatare obtained in the first step of the Quine-McCluskey algorithm starting from CDNFs. The completeminimization of DNFs of propositional logic, however, is an ambiguous problem in general. This can bedemonstrated by considering the following example of propositional logic:

P ∧ ¬Q ∨ ¬P ∧Q ∨ P ∧R ∨Q ∧R (14)

P ∧ ¬Q ∨ ¬P ∧Q ∨ P ∧R (15)

P ∧ ¬Q ∨ ¬P ∧Q ∨Q ∧R (16)

(15) and (16) are two equivalent completely minimized DNFs of the RDNF given in (14); cf. Quine [14]. Cf.also Kim [9, p. 67], who discusses the problems with identifying causally relevant factors based on minimizedDNFs.

15

– one single primary formula may imply another single primary formula or may besubcontrary or contrary to another single primary formula, and moreover, the sameapplies to truth functions of primary formulas (diversity of logical relations);

– one conjunction/disjunction of primary formulas may imply another conjunction/disjunctionof primary formulas, where all of the individual primary formulas are logically inde-pendent of each other (non-reducibility of logical relations to logical relations betweensingle primary formulas); and

– not only may conjunctions and disjunctions of FOLDNFs be equivalent to propercomponents thereof, but also

- primary formulas may be equivalent to primary formulas with fewer conjuncts ordisjuncts (atomic minimization), and

- a conjunction/disjunction of primary formulas may be equivalent to a conjunc-tion / disjunction of minimized primary formulas (context sensitivity of mini-mization).14

Thus, in the context of FOLDNFs, problems arise that are not considered in the Quine-McCluskey algorithm for minimizing propositional DNFs, such as the identification of incon-sistent primary formulas or inconsistent disjuncts and the minimization of primary formulas,disjuncts or disjunctions, because of relations of implication between primary formulas.

Furthermore, extending the two steps of the Quine-McCluskey algorithm to FOLDNFswould require more general and far more complex strategies. The first step of minimizingCDNFs of propositional logic in the Quine-McCluskey algorithm consists of merging dis-juncts following the merging rule A ∧ B ∨ A ∧ ¬B a` A (where B is atomic and A is aconjunction of length ≥ 1). In FOL, the merging of disjuncts is possible via a more generalrule: Given that a conjunction A1 implies a conjunction A2 and that the primary formulasB1 and B2 are subcontrary, then A1 ∧ B1 ∨ A2 ∧ B2 a` A1 ∨ A2 ∧ B2. Thus, eliminat-ing primary formulas within disjuncts of FOLDNFs based on internal relations with otherdisjuncts is a more complex task than the corresponding task in propositional logic. Thesame applies to eliminating components of a primary formula following some merging pro-cess (i.e., a process for minimizing disjuncts based on their relations with other disjuncts).Similar considerations apply to the second step of the Quine-McCluskey algorithm, whichminimizes disjunctions (instead of disjuncts, as in step 1). Minimizing a disjunction withinan FOLDNF based on its equivalence to a proper component of itself would require far morecomplex strategies than those adopted in the second step of the Quine-McCluskey algorithm.

Finally, a calculus for identifying internal relations (particularly relations of implication)within FOLDNFs is more complex than in the case of propositional logic because the relevantlogical implications also rest on special relations between quantifiers and their relations tovariables they bind. In particular, it is impossible to identify logical relations based on acomplete calculus without involving at least one rule that increases the complexity of theformulas in question; cf. p. 19 below.

14For example:

∃y1(Fy1 ∧ ∀x1(Gx1 ∨Hx1y1)) ∧ ∀x1Gx1 a` ∃y1Fy1 ∧ ∀x1Gx1 (17)

16

Compared with the case of general FOLDNFs, minimizing DNFs in propositional logic isa very special case and a trivial process. However, the fact that both theoretical and practicalarguments suggest that it is impossible to implement a general algorithm for minimizingFOLDNFs to the maximal extent does not mean that one should not seek a simplifiedalgorithm that would allow one to minimize arbitrary FOLDNFs to some extent. On theone hand, we must admit that we do not have a full understanding of the truth conditions of aformula φ unless we are able to provide a completely minimized FOLDNF of φ. On the otherhand, converting formulas into FOLDNFs and purging their (still unnecessarily intricate)representation of truth conditions of superfluous redundancies enhances the understandingof logical formulas, the more so as it enables the identification of the features that arerelevant for specifying purified truth conditions. Thus, if we wish to explain logical formulasand what they contribute to the representation of truth conditions, then there is no otherway but to provide minimized FOLDNFs, and the best we can do is to develop an algorithmthat helps to do so in a reasonably pragmatic way.

In the following, I informally outline an algorithm that I have implemented to mini-mize FOLDNFs. On the one hand, this algorithm should be helpful in cases in which ourunderstanding of FOL formulas is truly in need of help. This is why I do not confine thealgorithm to fragments of FOL for which a complete minimization, or at least decidability, isavailable, such as monadic FOL or formulas that do not contain quantifiers with scopes con-taining dyadic sentential connectives. Instead, given a complexity with which even trainedlogicians are unable to easily cope, the algorithm should provide a minimized FOLDNF foran arbitrary formula φ of pure FOL. On the other hand, the algorithm should be suitablyeffective and provide outputs for formulas of some complexity in a reasonable amount oftime. Admittedly, when the objective is conversion into FOLDNFs and the subsequent min-imization of the result, initial formulas of great length (very roughly speaking, formulas ofmore than approximately 5 lines) that contain many internal relations (roughly speaking,formulas containing many occurrences of the same propositional function) are beyond whata sufficiently powerful algorithm can process in a reasonable amount of time. However,supposing that the initial formula is too complex to be grasped by a trained human but ofsufficiently low complexity to be manageable by a computer, then the algorithm should pro-vide a minimized FOLDNF within a reasonable amount of time (say, in most cases, withinless than 10 minutes on a normal machine).

To this end, the algorithm I suggest restricts minimization according to the followingrules:

1. It restricts minimization to

(a) minimization of single primary formulas,

(b) minimization of conjunctions of primary formulas based on internal relations be-tween pairs of primary formulas within a given conjunction, and

(c) minimization of disjunctions of conjunctions of primary formulas based on in-ternal relations between primary formulas of pairs of disjuncts within a givendisjunction.

In particular, the algorithm does not provide any equivalent of the two steps of theQuine-McCluskey algorithm; it abstains from defining any equivalent for the intricate

17

process of merging as well as from adopting strategies for identifying, in general, theequivalence of conjunctions or disjunctions to any of their proper components.

2. It makes use of a restricted, incomplete calculus to identify relations of implicationbetween primary formulas.

3. It makes use of time constraint commands.

Item 1 above is explained in section 4.3; items 2 and 3 are explained in the followingsection.

4.2 Derivation Trees

Given the syntax of primary formulas, a correct and complete calculus identifying relationsof implication between primary formulas can be defined by considering the minimal syntac-tic differences of the significant syntactic features of FOLDNFs. Table 6 provides a briefoverview of the 13 rules of the calculus that specify those minimal syntactic differences be-tween FOLDNFs that are truth-preserving. In fact, these rules are correct for any NNFs.Each rule concerns only a minimal syntactic difference between NNFs. The rules applyregardless of where the syntactic feature occurs within the formula. “E” denotes “elimina-tion”. “I” stands for “introduction”. “µ/ν” means that the variable µ is replaced with avariable ν, which is new if ν does not appear to the left of “`”. “µ, µ/ν” means that some(not necessarily all) of the occurrences of µ are replaced with ν.

∃∀Ex: ∃µ∀ν ` ∀ν∃µ∀E1/∃I1: ∀µA(µ) ` ∃νA(µ/ν)

∀E2: ∀µ∀νA(µ, ν) ` ∀µA(µ, ν/µ) ∃I2: ∃µA(µ) ` ∃µ∃νA(µ, µ/ν)

∀E3: ∃µ∀νA(µ, ν) ` ∃µA(ν/µ) ∃I3: ∀µA(µ) ` ∀µ∃νA(µ, µ/ν)

∨E: A ∨A ` A ∨I: A ` A ∨B∧E: A ∧B ` A ∧I: A ` A ∧A⊥E: A ∨ ⊥ ` A >I: A ` A ∧ >

Table 6: Rules of implication

In fact, the algorithm does not make use of ∨I, ∧I or >I at all, and it applies theremaining rules only to identify relations of implication between primary formulas. To doso, sequences of existential quantifiers are defined as “orderless”, i.e., the rules apply toany existential quantifier in the sequence. Existential quantifiers that are separated onlyby disjunctions or conjunctions are treated as existential quantifiers of one and the samesequence. Thus, only existential quantifiers separated by universal quantifiers are consideredto belong to different sequences. The same applies to sequences of universal quantifiers.Thus, strictly speaking, the rules are also applied to conjunctions or disjunctions of primaryformulas if those primary formulas are preceded by quantifiers of the same sort. It is always

18

possible to group such quantifiers into one sequence by means of PN laws. Conjunctions anddisjunctions are similarly orderless, and ∧E, ∨E and ⊥E are also applied to conjunctionsof primary formulas or to disjunctions of conjunctions of primary formulas. The resultingformula on the right-hand side of “`” need not be a primary formula (or an FOLDNF). If itis not, it must be converted into a primary formula (or an FOLDNF) through equivalencetransformation. This may result in conjunctions or disjunctions of primary formulas, towhich the rules may, in turn, be applied. In this way, primary formulas that are impliedby other primary formulas can be identified through iterative application of the rules viaconjunctions/disjunctions of primary formulas.

The correctness of the calculus can be proven either by paraphrasing or by translationinto well-known logical rules applying to NNFs. The completeness of the rules can beproven by considering all (finite) additional possible minimal syntactic differences betweenFOLDNFs and by demonstrating, either by paraphrasing or based on model theory, thatthey are not truth-preserving. However, as we do not claim that the minimization algorithmis complete, we need not go into detail here.

Instead, the algorithm makes use of derivation trees, starting from primary formulas anditeratively applying only 10 of the 13 rules of implication in all possible ways (cf. figure 1for a rather simple example). The three rules ∨I, ∧I and >I all increase the complexityof the formula and are thus omitted to ensure the generation of derivation trees of finitesize. In addition, a time constraint command is used to restrict the generation of thederivation trees in complex cases. The time required to minimize an FOLDNF essentiallydepends on the number of derivation trees that are generated, which, in turn, depends onthe number of primary formulas of the FOLDNF. As this number is finite and the generatedderivation trees are finite in size, the minimization procedure is guaranteed to terminate.Time constraint commands ensure that the procedure terminates in a reasonable amount oftime given FOLDNFs of a reasonable complexity.

Whereas one can compensate for not using ∨I by applying refined rules for identifyingwhether one primary formula is implied by another (cf. p. 22), refraining from using ∧I and>I makes it impossible to identify relations of implication between certain primary formulas.A single example may suffice to demonstrate this. The following formula is a conjunction oftwo primary formulas:

∃y1(¬Fy1 ∧ ∃y2(Fy2 ∧ ∃y3(¬Gy1y3 ∧ ¬Gy3y2))) ∧∀x1(Fx1 ∨ ∀x2(¬Fx2 ∨Gx1x2)) (18)

This conjunction is inconsistent, as shown by the fact that the first primary formula impliesthe negation of the second (converted into a primary formula, i.e., the formula in line (5) oftable 7). However, to prove this, one must apply either ∧I or >I in addition to some of the10 rules used for the construction of derivation trees. In the derivation presented in table 7,>I is applied.

In addition, ∀E3 is applied to replace x1 with y3 in the formula appearing in line (2),which is possible because ∃y1∃y2∃y3 is an orderless sequence by definition. The result of∀E3 is transformed into a disjunction of primary formulas in line (3). ∧E is then applied toeliminate ¬Fy1 and ¬Gy1y3 in the first disjunct of the formula in line (3) and to eliminateFy2 and ¬Gy3y2 in the second disjunct. Finally, ∨E eliminates one of the two identicaldisjuncts in line (4). The variables are renamed to standardize the variable usage in each

19

∀x1∀x2Fx1x2��

��

���

∀E2�

��

����+

∀x1Fx1x1

∀E1/∃I1

?∃y1∀x1Fy1x1

QQQQQQQ∀E1/∃I1

QQQQQQQs∃y1∀x1Fx1y1

∀E1/∃I1

?

QQQQ∃I3QQQQQQQQQQs

PPPPPPPPPPPPPPPPPPPPPP∃I3 PPPPPPPPq

��

��∀E3�

��

��

���

��+

��������∀E3����������������������)

∃∀Ex

?

∃∀Ex

?∃y1Fy1y1 ∀x1∃y1Fy1x1 ∀x1∃y1Fx1y1

QQQQQQQ∃I2QQQQQQQs

∀EI/∃I1

?

��

���

��∀E1/∃I1�

���

���+

∃y1∃y2Fy1y2

Figure 1: Derivation tree starting from ∀x1x2Fx1x2

(1) ∃y1(¬Fy1 ∧ ∃y2(Fy2 ∧ ∃y3(¬Gy1y3 ∧ ¬Gy3y2))) (18)

(2) ∃y1(¬Fy1 ∧ ∃y2(Fy2 ∧ ∃y3(¬Gy1y3 ∧ ¬Gy3y2))) ∧ ∀x1(Fx1 ∨ ¬Fx1) >I

(3)∃y1(¬Fy1 ∧ ∃y2(Fy2 ∧ ∃y3(¬Gy1y3 ∧ ¬Gy3y2 ∧ Fy3))) ∨∃y1(¬Fy1 ∧ ∃y2(Fy2 ∧ ∃y3(¬Gy1y3 ∧ ¬Gy3y2 ∧ ¬Fy3)))

∀E3

(4)∃y1(¬Fy1 ∧ ∃y2(Fy2 ∧ ¬Gy1y2)) ∨∃y1(¬Fy1 ∧ ∃y2(Fy2 ∧ ¬Gy1y2))

4× ∧E

(5) ∃y1(¬Fy1 ∧ ∃y2(Fy2 ∧ ¬Gy1y2)) ∨E

Table 7: Derivation using >I

20

case.Thus, abstaining from the application of ∧I and >I implies incompleteness. Conse-

quently, using this approach prevents complete minimization. However, if one were to per-mit the iterative application of ∧I or >I, then the size of the derivation trees would increaseto infinity. By relying on the derivation trees generated by all possible applications of theremaining 10 of the 13 rules listed in table 6, the minimization strategy achieves terminationat the expense of complete minimization.

4.3 FOL Optimizer

I call the program implemented to generate a minimized FOLDNF for a given formula φ ofpure FOL “FOL Optimizer”. Roughly speaking, the program proceeds as follows (cf. figure2):

1. Transform φ into an FOLDNF using the algorithm described in section 2.

2. Minimize single primary formulas:

(a) Eliminate redundant conjuncts within primary formulas.

(b) Eliminate redundant disjuncts within primary formulas.

(c) Eliminate an entire primary formula if its negation can be identified as inconsis-tent.15

3. Minimize single disjuncts of the result obtained in step 2:

(a) Eliminate an entire disjunct if some primary formula thereof can be identified asinconsistent.

(b) Eliminate an entire disjunct if some primary formula thereof implies the negationof some other primary formula thereof.16

(c) Eliminate a single primary formula if it is implied by some other primary formulaof the same disjunct.

4. Minimize the resulting disjunction obtained in step 3:

(a) If two disjuncts of length 1 contain primary formulas that can be identified assubcontrary (i.e., the negation of primary formula A implies primary formula B),then return a result of “True” (indicating a tautology).

(b) Eliminate a disjunct D1 if all primary formulas of another disjunct D2 are impliedby the primary formulas of D1.

The algorithm returns a result of “False” in the case that φ can be identified as incon-sistent and a result of “True” in the case that φ can be identified as tautologous; otherwise,it returns a minimized FOLDNF that is equivalent to the initial formula φ.

15If all primary formulas of a disjunct of the FOLDNF are thus identified as tautologous, then φ is identifiedas tautologous.

16If all disjuncts of the FOLDNF are identified as inconsistent, then φ is identified as inconsistent.

21

Step 1, in fact, also involves the application of well-known propositional minimizationprocedures (beyond the algorithm described in section 2). These minimization proceduresand those applied in steps 2 to 4 all minimize conjuncts or disjuncts within FOLDNFs (andthus NNFs) and rely on the following equivalence rules (from left to right):

A ∧ A a` A IP1a∧A ∨ A a` A IP1a∨(A ∨B) ∧ (A ∨ ¬B) a` A IP2∧(A ∧B) ∨ (A ∧ ¬B) a` A IP2∨> ∧ A a` A IP1>> ∨ A a` > IP2>⊥ ∧A a` ⊥ IP1⊥⊥ ∨A a` A IP2⊥A ∧ B a` A if A ` B IP1b∧A ∨ B a` A if B ` A IP1b∨Table 8: Equivalence rules used for minimization

The last two rules are used only in steps 2 to 4 of the algorithm. Whether the conditionsA ` B and B ` A for these two rules hold is decided by referring to derivations trees;cf. section 4.2. Reference to derivation trees is also made in steps 2 to 4 to identify theconditions > and ⊥ for IP1/2> and IP1/2⊥. To apply all stated minimization rules to themaximal extent, minimal renaming (cf. p. 8) is used.

In addition to the stated rules, we tacitly assume the following equivalence rules:

A ∨ ¬A a` > R1A ∧ ¬A a` ⊥ R2> a` ¬ ⊥ R3A ∧B a` ⊥ if A ` ¬B R4A ∨B a` > if ¬A ` B R5

Table 9: Tacitly assumed equivalence rules

In step 2(a), redundant conjuncts within a primary formula A are identified by substi-tuting single conjuncts of a conjunction C with other single conjuncts of C. Each suchsubstitution results in a primary formula A′. For each such formula A′, a derivation tree isgenerated. Furthermore, from the primary formula A, all possible formulas are generatedsuch that disjuncts within A are deleted. The resulting set of formulas, including A, isdenoted by A∗. The set A∗ is generated because ∨I is not applied for the generation ofderivation trees. However, A′ implies A if A′ implies some formula in the set A∗, because allformulas in A∗ imply A. As a substitute for deriving A from A′ by using ∨I, the formulasin A∗ can be derived from A′ without using ∨I. This “trick” for circumventing the use of∨I is also used in all other steps of the algorithm when the objective is to test whethersome primary formula is implied by another primary formula using derivation trees. Fur-thermore, the variables are always standardized by means of minimal renaming; cf. p. 8.This enables the algorithm to decide whether some primary formula X is implied by another

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φ

?

FOLDNF

?

Minimization 1 (primary formulas):(a). conjuncts via IP1b∧(b). disjuncts via IP1b∨(c). entire primary formula via IP>1 + R3

?

Minimization 2 (disjuncts):(a). entire disjunct via IP2⊥(b). entire disjunct via IP2⊥ + R4(c). primary formula via IP1b∨

?

Minimization 3 (disjunction):(a). entire disjunction via IP2> + R5(b). disjuncts via IP1b∨

Figure 2: Chart of the algorithm

primary formula Y by virtue of the 10 rules applied to generate a derivation tree for Y bychecking whether X is a member of that tree. Redundant conjuncts are thus identified bychecking whether some (standardized) formula of A∗ is a member of the derivation tree ofA′. If a conjunct is redundant, it is eliminated; the resulting formula is transformed into aprimary formula through equivalence transformation, and its variables are standardized bymeans of minimal renaming. This process is iteratively applied for all single conjuncts of allconjunctions of all primary formulas. To limit the construction of derivation trees, trees aregenerated only if the conjunct in question and the conjunct that is substituted for it containat least one literal of the same length, involving the same predicate letter and preceded bya negation sign in either both cases or neither. This is a trivially necessary condition forthe redundancy of a conjunct based on its internal relation to another conjunct in the sameconjunction. Furthermore, the generation of derivation trees terminates as soon as somemember of A∗ is a member of the tree.17

Step 2(b) proceeds in a manner analogous to that of step 2(a). Because disjuncts insteadof conjuncts are being considered, a derivation tree is generated for A, and the algorithmtests whether some formula of the set of formulas A′∗ is a member of that tree.

17In addition, the generation of derivation trees is always restricted by a time constraint of 10 seconds,which is sufficient unless the primary formulas are quite complex. The generation of derivation trees alsoterminates if an explicit contradiction (identifiable based on propositional contradictions of quantifier scopes)is derived. In this case, the corresponding primary formula, and thus the disjunct that contains it, isidentified as inconsistent. Consequently, the inconsistent disjunct is deleted unless it is the only disjunct ofthe FOLDNF, in which case the initial formula φ is identified as inconsistent.

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In step 2(c), tautologous primary formulas are identified by generating the derivationtrees for their negations converted into standardized primary formulas. If some explicitcontradictory formula can be identified within a derivation tree, then the primary formulaunder consideration is tautologous. Several plain necessary conditions for inconsistency areimplemented to limit the generation of derivation trees.

In step 3, derivation trees of primary formulas are generated from single disjuncts ofthe FOLDNF obtained in step 2. As soon as a derivation tree contains an explicit contra-diction (step 3(a)) or the negation of some other primary formula, where this negation isstandardized and converted into a primary formula (step 3(b)), the corresponding disjunctis identified as inconsistent and is deleted. Otherwise, a primary formula A is deleted ifsome formula of the set A∗ is a member of some derivation tree of another primary formulaof the same disjunct (step 3(c)).

In step 4, derivation trees of primary formulas of the resulting FOLDNF obtained in step3 are generated to identify internal relations between primary formulas of different disjuncts.Step 4(a) involves a simple test for tautologies with respect to relations of implication be-tween single primary formulas of different disjuncts. If the derivation tree of the negation ofa primary formula A (standardized and transformed into a primary formula) contains someformula in the set A′∗ of a primary formula A′, where A and A′ are distinct disjuncts of theFOLDNF resulting from step 3, then the initial formula φ is a tautology. In step 4(b), adisjunct D1 is deleted if each set A′∗ of primary formulas of another disjunct D2 containssome member that is a member of the union of the derivation trees of the primary formulasof D1. Again, several plain criteria are implemented to restrict the generation of derivationtrees in step 4.

The algorithm can be trivially seen to terminate because (i) step 1 (transformationinto an FOLDNF) terminates (cf. p. 8) and (ii) steps 2-4 successively minimize finiteparts of the FOLDNF according to decisions regarding the implication relations betweenprimary formulas, which terminate because of time constraint commands and the restrictionto (incomplete) derivation trees of finite size (cf. p. 19). The algorithm is correct, asit involves the application of nothing but logical equivalence rules. Finally, it serves itsexpected purpose (namely, minimizes an FOLDNF to some extent), as it first transformsan initial formula φ into an FOLDNF and then uses logical equivalence rules to minimizethat FOLDNF by eliminating conjuncts and/or disjuncts.

Figure 3 illustrates the results of the various steps of the algorithm for the followingexample test:

¬∀y6¬¬∀y3¬∃y2∃y4∀x1¬(¬Fy3y3 ∨ ¬(∀x2(∃y1Fy2y1 ∧(¬Fy2y4 → Fy3x1) ∧ ∃y5∃y7((Fy5y7 ∧ Fy5x2) ∨ Fy6x2)))) test

The truth conditions of instances of test are difficult to grasp. A literal paraphrase oftest provides no assistance. Transforming test into an FOLDNF according to step 1 of theFOL Optimizer procedure is of some help. However, the result contains many redundanciesthat still obscure the truth conditions. Subsequent stepwise minimization makes the truthconditions increasingly clear. The result, ∀x1∃y1Fx1y1 (or T − {{∀1∃1T − F12}} in iconicnotation), finally explains the truth conditions for the initial formula test.

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Figure 3: Optimization of test using FOL Optimizer

Acknowledgement

I am grateful to Markus Sabel for comments on a previous version of this paper. I wouldalso like to thank the anonymous referees for their helpful suggestions.

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